In this section, we conduct various simulation studies to assess the performance of the proposed methods, TPST and ATPST. For the ATPST, we consider the weight functions defined in Section 3.2 with \(\tau = 2\)  and \(C\)  being selected from a set of grid points ranging from 1 to 3 based on the GCV criteria. We compare the proposed methods with the traditional tensor product spline method [1]} and thin plate spline smoothing [2]}. We then implement the tensor product spline and the thin plate spline smoothing using the function gam in R package mgcv.
